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(a^2+a-2)x^2+(2a^2+a+3)x+a^2-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
/ 2 \ 2 / 2 \ 2 \a + a - 2/*x + \2*a + a + 3/*x + a - 1 = 0
$$\left(a^{2} + \left(x^{2} \left(\left(a^{2} + a\right) - 2\right) + x \left(\left(2 a^{2} + a\right) + 3\right)\right)\right) - 1 = 0$$
Solución detallada
Abramos la expresión en la ecuación
$$\left(a^{2} + \left(x^{2} \left(\left(a^{2} + a\right) - 2\right) + x \left(\left(2 a^{2} + a\right) + 3\right)\right)\right) - 1 = 0$$
Obtenemos la ecuación cuadrática
$$a^{2} x^{2} + 2 a^{2} x + a^{2} + a x^{2} + a x - 2 x^{2} + 3 x - 1 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = a^{2} + a - 2$$
$$b = 2 a^{2} + a + 3$$
$$c = a^{2} - 1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{- 2 a^{2} - a + \sqrt{- \left(a^{2} - 1\right) \left(4 a^{2} + 4 a - 8\right) + \left(2 a^{2} + a + 3\right)^{2}} - 3}{2 a^{2} + 2 a - 4}$$
$$x_{2} = \frac{- 2 a^{2} - a - \sqrt{- \left(a^{2} - 1\right) \left(4 a^{2} + 4 a - 8\right) + \left(2 a^{2} + a + 3\right)^{2}} - 3}{2 a^{2} + 2 a - 4}$$
$$\left(a^{2} + \left(x^{2} \left(\left(a^{2} + a\right) - 2\right) + x \left(\left(2 a^{2} + a\right) + 3\right)\right)\right) - 1 = 0$$
Obtenemos la ecuación cuadrática
$$a^{2} x^{2} + 2 a^{2} x + a^{2} + a x^{2} + a x - 2 x^{2} + 3 x - 1 = 0$$
Es la ecuación de la forma
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = a^{2} + a - 2$$
$$b = 2 a^{2} + a + 3$$
$$c = a^{2} - 1$$
, entonces
D = b^2 - 4 * a * c =
(3 + a + 2*a^2)^2 - 4 * (-2 + a + a^2) * (-1 + a^2) = (3 + a + 2*a^2)^2 - (-1 + a^2)*(-8 + 4*a + 4*a^2)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{- 2 a^{2} - a + \sqrt{- \left(a^{2} - 1\right) \left(4 a^{2} + 4 a - 8\right) + \left(2 a^{2} + a + 3\right)^{2}} - 3}{2 a^{2} + 2 a - 4}$$
$$x_{2} = \frac{- 2 a^{2} - a - \sqrt{- \left(a^{2} - 1\right) \left(4 a^{2} + 4 a - 8\right) + \left(2 a^{2} + a + 3\right)^{2}} - 3}{2 a^{2} + 2 a - 4}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a^{2} + x^{2} \left(a^{2} + a - 2\right) + x \left(2 a^{2} + a + 3\right) - 1 = 0$$
Коэффициент при x равен
$$a^{2} + a - 2$$
entonces son posibles los casos para a :
$$a < -2$$
$$a = -2$$
$$a > -2 \wedge a < 1$$
$$a = 1$$
Consideremos todos los casos con detalles:
Con
$$a < -2$$
la ecuación será
$$4 x^{2} + 18 x + 8 = 0$$
su solución
$$x = -4$$
$$x = - \frac{1}{2}$$
Con
$$a = -2$$
la ecuación será
$$9 x + 3 = 0$$
su solución
$$x = - \frac{1}{3}$$
Con
$$a > -2 \wedge a < 1$$
la ecuación será
$$- \frac{9 x^{2}}{4} + 3 x - \frac{3}{4} = 0$$
su solución
$$x = \frac{1}{3}$$
$$x = 1$$
Con
$$a = 1$$
la ecuación será
$$6 x = 0$$
su solución
$$x = 0$$
$$a^{2} + x^{2} \left(a^{2} + a - 2\right) + x \left(2 a^{2} + a + 3\right) - 1 = 0$$
Коэффициент при x равен
$$a^{2} + a - 2$$
entonces son posibles los casos para a :
$$a < -2$$
$$a = -2$$
$$a > -2 \wedge a < 1$$
$$a = 1$$
Consideremos todos los casos con detalles:
Con
$$a < -2$$
la ecuación será
$$4 x^{2} + 18 x + 8 = 0$$
su solución
$$x = -4$$
$$x = - \frac{1}{2}$$
Con
$$a = -2$$
la ecuación será
$$9 x + 3 = 0$$
su solución
$$x = - \frac{1}{3}$$
Con
$$a > -2 \wedge a < 1$$
la ecuación será
$$- \frac{9 x^{2}}{4} + 3 x - \frac{3}{4} = 0$$
su solución
$$x = \frac{1}{3}$$
$$x = 1$$
Con
$$a = 1$$
la ecuación será
$$6 x = 0$$
su solución
$$x = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(a^{2} + \left(x^{2} \left(\left(a^{2} + a\right) - 2\right) + x \left(\left(2 a^{2} + a\right) + 3\right)\right)\right) - 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a^{2} + x^{2} \left(a^{2} + a - 2\right) + x \left(2 a^{2} + a + 3\right) - 1}{a^{2} + a - 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2 a^{2} + a + 3}{a^{2} + a - 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a^{2} - 1}{a^{2} + a - 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2 a^{2} + a + 3}{a^{2} + a - 2}$$
$$x_{1} x_{2} = \frac{a^{2} - 1}{a^{2} + a - 2}$$
$$\left(a^{2} + \left(x^{2} \left(\left(a^{2} + a\right) - 2\right) + x \left(\left(2 a^{2} + a\right) + 3\right)\right)\right) - 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a^{2} + x^{2} \left(a^{2} + a - 2\right) + x \left(2 a^{2} + a + 3\right) - 1}{a^{2} + a - 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2 a^{2} + a + 3}{a^{2} + a - 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a^{2} - 1}{a^{2} + a - 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2 a^{2} + a + 3}{a^{2} + a - 2}$$
$$x_{1} x_{2} = \frac{a^{2} - 1}{a^{2} + a - 2}$$
Gráfica
Suma y producto de raíces
[src]
suma
2 2 / (1 - re(a))*im(a) (2 + re(a))*im(a) \ im (a) (1 - re(a))*(2 + re(a)) / (1 + re(a))*im(a) (-1 + re(a))*im(a) \ im (a) (1 + re(a))*(-1 + re(a)) I*|- --------------------- - ---------------------| - --------------------- + ----------------------- + I*|---------------------- - ----------------------| - ---------------------- - ------------------------ | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 \ (2 + re(a)) + im (a) (2 + re(a)) + im (a)/ (2 + re(a)) + im (a) (2 + re(a)) + im (a) \(-1 + re(a)) + im (a) (-1 + re(a)) + im (a)/ (-1 + re(a)) + im (a) (-1 + re(a)) + im (a)
$$\left(i \left(- \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(a\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \left(\frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
2 2 / (1 + re(a))*im(a) (-1 + re(a))*im(a) \ / (1 - re(a))*im(a) (2 + re(a))*im(a) \ im (a) im (a) (1 - re(a))*(2 + re(a)) (1 + re(a))*(-1 + re(a)) I*|---------------------- - ----------------------| + I*|- --------------------- - ---------------------| - ---------------------- - --------------------- + ----------------------- - ------------------------ | 2 2 2 2 | | 2 2 2 2 | 2 2 2 2 2 2 2 2 \(-1 + re(a)) + im (a) (-1 + re(a)) + im (a)/ \ (2 + re(a)) + im (a) (2 + re(a)) + im (a)/ (-1 + re(a)) + im (a) (2 + re(a)) + im (a) (2 + re(a)) + im (a) (-1 + re(a)) + im (a)
$$\frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + i \left(- \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(a\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
producto
/ 2 \ / 2 \ | / (1 - re(a))*im(a) (2 + re(a))*im(a) \ im (a) (1 - re(a))*(2 + re(a))| | / (1 + re(a))*im(a) (-1 + re(a))*im(a) \ im (a) (1 + re(a))*(-1 + re(a))| |I*|- --------------------- - ---------------------| - --------------------- + -----------------------|*|I*|---------------------- - ----------------------| - ---------------------- - ------------------------| | | 2 2 2 2 | 2 2 2 2 | | | 2 2 2 2 | 2 2 2 2 | \ \ (2 + re(a)) + im (a) (2 + re(a)) + im (a)/ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / \ \(-1 + re(a)) + im (a) (-1 + re(a)) + im (a)/ (-1 + re(a)) + im (a) (-1 + re(a)) + im (a) /
$$\left(i \left(- \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(a\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) \left(\frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
2 2
2 + im (a) + re (a) + 3*re(a) + I*im(a)
---------------------------------------
2 2
4 + im (a) + re (a) + 4*re(a)
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2} + 3 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + i \operatorname{im}{\left(a\right)} + 2}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 4}$$
(2 + im(a)^2 + re(a)^2 + 3*re(a) + i*im(a))/(4 + im(a)^2 + re(a)^2 + 4*re(a))
Respuesta rápida
[src]
2
/ (1 - re(a))*im(a) (2 + re(a))*im(a) \ im (a) (1 - re(a))*(2 + re(a))
x1 = I*|- --------------------- - ---------------------| - --------------------- + -----------------------
| 2 2 2 2 | 2 2 2 2
\ (2 + re(a)) + im (a) (2 + re(a)) + im (a)/ (2 + re(a)) + im (a) (2 + re(a)) + im (a)
$$x_{1} = \frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(1 - \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
2
/ (1 + re(a))*im(a) (-1 + re(a))*im(a) \ im (a) (1 + re(a))*(-1 + re(a))
x2 = I*|---------------------- - ----------------------| - ---------------------- - ------------------------
| 2 2 2 2 | 2 2 2 2
\(-1 + re(a)) + im (a) (-1 + re(a)) + im (a)/ (-1 + re(a)) + im (a) (-1 + re(a)) + im (a)
$$x_{2} = i \left(- \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\left(\operatorname{re}{\left(a\right)} - 1\right) \left(\operatorname{re}{\left(a\right)} + 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x2 = i*(-(re(a) - 1)*im(a)/((re(a) - 1)^2 + im(a)^2) + (re(a) + 1)*im(a)/((re(a) - 1)^2 + im(a)^2)) - (re(a) - 1)*(re(a) + 1)/((re(a) - 1)^2 + im(a)^2) - im(a)^2/((re(a) - 1)^2 + im(a)^2)